Probability Statistics Practice Questions

Generate GATE-level questions covering probability axioms, conditional probability, Bayes' theorem, random variables (discrete/continuous), probability distributions (binomial, Poisson, normal, exponential), expectation, variance, correlation, regression, and sampling methods.

1 questions · 1 PYQs · 0 AI practice · GATE CSE 2027

📘 Probability Statistics – Concept Summary

### 1. Definition Probability: Mathematical framework for uncertainty. Statistics: Methods for collecting, analyzing, and interpreting data. ### 2. Basic Probability (Very Important) - P(A∪B) = P(A) + P(B) − P(A∩B) - P(A|B) = P(A∩B) / P(B) [Conditional Probability] - A and B independent iff P(A∩B) = P(A)·P(B) - P(A') = 1 − P(A) **Bayes' Theorem**: P(A|B) = P(B|A)·P(A) / P(B) = P(B|A)·P(A) / [P(B|A)·P(A) + P(B|A')·P(A')] **Total Probability**: P(B) = Σ P(B|Aᵢ)·P(Aᵢ) for partition {Aᵢ} ### 3. Random Variables - **Discrete RV**: Countable values. PMF: P(X=x) ≥ 0, ΣP(X=x) = 1 - **Continuous RV**: PDF f(x) ≥ 0, ∫f(x)dx = 1. P(a≤X≤b) = ∫ₐᵇ f(x)dx **Expectation & Variance**: - E[X] = Σ x·P(X=x) [discrete] or ∫x·f(x)dx [continuous] - Var(X) = E[X²] − (E[X])² - E[aX+b] = aE[X]+b - Var(aX+b) = a²Var(X) ### 4. Standard Distributions (Very Important) **Discrete**: | Distribution | PMF | Mean | Variance | |---|---|---|---| | Bernoulli(p) | P(X=1)=p | p | p(1−p) | | Binomial(n,p) | ⁿCₓ pˣ(1−p)ⁿ⁻ˣ | np | np(1−p) | | Poisson(λ) | e⁻λλˣ/x! | λ | λ | | Geometric(p) | (1−p)ˣ⁻¹p | 1/p | (1−p)/p² | **Continuous**: | Distribution | PDF | Mean | Variance | |---|---|---|---| | Uniform(a,b) | 1/(b−a) | (a+b)/2 | (b−a)²/12 | | Exponential(λ) | λe⁻λx | 1/λ | 1/λ² | | Normal(μ,σ²) | Bell curve | μ | σ² | ### 5. Normal Distribution - Standard normal: Z = (X−μ)/σ ~ N(0,1) - 68-95-99.7 rule: P(μ±σ)=68%, P(μ±2σ)=95%, P(μ±3σ)=99.7% - Sum of independent normals: X+Y ~ N(μ₁+μ₂, σ₁²+σ₂²) ### 6. Statistics **Central Tendency**: Mean = Σxᵢ/n, Median = middle value, Mode = most frequent **Dispersion**: Variance = Σ(xᵢ−x̄)²/n, SD = √Variance **Covariance & Correlation**: - Cov(X,Y) = E[XY] − E[X]E[Y] - ρ = Cov(X,Y) / (σₓ·σᵧ), −1 ≤ ρ ≤ 1 - Independent → Cov = 0, but Cov = 0 does NOT imply independence ### 7. Conditional Expectation E[X+Y] = E[X] + E[Y] always (linearity) E[XY] = E[X]·E[Y] only if X,Y independent ### 8. GATE Trick Poisson: both mean and variance = λ (unique property). Exponential distribution is memoryless: P(X>s+t|X>s) = P(X>t). Bayes' theorem questions: carefully identify prior, likelihood, and evidence. Binomial → Poisson approximation when n large, p small, np=λ moderate.

Q1GATE 2019NAT

Two numbers are chosen independently and uniformly at random from the set {1, 2, ..., 13}. The probability (rounded off to 3 decimal places) that their 4-bit (unsigned) binary representations have the same most significant bit is ________.

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📘 Probability Statistics – Concept Summary

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